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  "signature": "sha256:1ea7eb5baa9b54b9c8bb4ec97fe085e1ae285d5775adecea239fa6b380a18a68"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Simplification"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "x, y, z = symbols('x y z')\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For each exercise, fill in the function according to its docstring."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Polynomial/Rational Function Simplification"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In each exercise, apply specific simplification functions to get the desired result."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def polysimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> polysimp1(cos(x)*sin(x) + cos(x))\n",
      "    (sin(x) + 1)*cos(x)\n",
      "    >>> polysimp1(cos(x)*sin(x) + cos(x) + 1)\n",
      "    (sin(x) + 1)*cos(x) + 1\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "polysimp1(cos(x)*sin(x) + cos(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "polysimp1(cos(x)*sin(x) + cos(x) + 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def polysimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> polysimp2((2*x + 1)/(x**2 + x))\n",
      "    1/(x + 1) + 1/x\n",
      "    >>> polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))\n",
      "    1/(x**2 + 2*x + 1) + 1/x\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "polysimp2((2*x + 1)/(x**2 + x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Powers"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In each exercise, apply specific simplification functions to get the desired result. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def powersimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> powersimp1(exp(x)*(exp(y) + 1))\n",
      "    exp(x) + exp(x + y)\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "powersimp1(exp(x)*(exp(y) + 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def powersimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> powersimp2(2**x*x**x)\n",
      "    (2*x)**x\n",
      "    >>> powersimp2(x**x*x**x)\n",
      "    (x**2)**x\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "powersimp2(2**x*x**x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "powersimp2(x**x*x**x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def powersimp3(expr):\n",
      "    \"\"\"\n",
      "    >>> a, b, c = symbols('a b c')\n",
      "    >>> powersimp3((a**b)**c)\n",
      "    a**(b*c)\n",
      "    >>> powersimp3((a**b)**(c + 1))\n",
      "    a**(b*c + b)\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b, c = symbols('a b c')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "powersimp3((a**b)**c)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "powersimp3((a**b)**(c + 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Logs"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def logsimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> a, b = symbols('a b', positive=True)\n",
      "    >>> logsimp1(log(x**y*a**b))\n",
      "    y*log(x) + log(a**b)\n",
      "    >>> logsimp1(log(x*y*a*b))\n",
      "    log(x) + log(y) + log(a*b)\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b = symbols('a b', positive=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "logsimp1(log(x**y*a**b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "logsimp1(log(x*y*a*b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Miscellaneous  "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def miscsimp1(expr):\n",
      "    \"\"\"\n",
      "    >>> miscsimp1(sin(x + y))\n",
      "    2*(-tan(x/2)**2 + 1)*tan(y/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1)) + 2*(-tan(y/2)**2 + 1)*tan(x/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1))\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "miscsimp1(sin(x + y))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def miscsimp2(expr):\n",
      "    \"\"\"\n",
      "    >>> miscsimp2(gamma(x + 4))\n",
      "    x**4*gamma(x) + 6*x**3*gamma(x) + 11*x**2*gamma(x) + 6*x*gamma(x)\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "miscsimp2(gamma(x + 4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Continued Fractions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we do not cover this, see http://asmeurer.github.io/scipy-2014-tutorial/html/tutorial/simplification.html#example-continued-fractions."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def list_to_frac(l):\n",
      "    expr = Integer(0)\n",
      "    for i in reversed(l[1:]):\n",
      "        expr += i\n",
      "        expr = 1/expr\n",
      "    return l[0] + expr"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a0, a1, a2, a3, a4 = symbols('a0:5')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Determine the list used to create the continued fraction $$\\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}.$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def continued_frac():\n",
      "    \"\"\"\n",
      "    Determine the original list used to create the fraction.  \n",
      "\n",
      "    Return the original list from this function.\n",
      "\n",
      "    >>> orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)\n",
      "    >>> pprint(orig_frac)\n",
      "    a\u2080\u22c5a\u2081\u22c5a\u2082\u22c5a\u2083\u22c5a\u2084 + a\u2080\u22c5a\u2081\u22c5a\u2082 + a\u2080\u22c5a\u2083\u22c5a\u2084 + a\u2080 + a\u2081\u22c5a\u2082\u22c5a\u2083 + a\u2081\u22c5a\u2083\u22c5a\u2084 + a\u2081 + a\u2083\n",
      "    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
      "                     a\u2080\u22c5a\u2081\u22c5a\u2082\u22c5a\u2084 + a\u2080\u22c5a\u2084 + a\u2081\u22c5a\u2082 + a\u2081\u22c5a\u2084 + 1\n",
      "    >>> cancel(list_to_frac(continued_frac())) == orig_frac\n",
      "    True\n",
      "    \"\"\"\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "orig_frac"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAkgAAAArBAMAAABxz2pBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 30,
       "text": [
        "a\u2080\u22c5a\u2081\u22c5a\u2082\u22c5a\u2083\u22c5a\u2084 + a\u2080\u22c5a\u2081\u22c5a\u2082 + a\u2080\u22c5a\u2083\u22c5a\u2084 + a\u2080 + a\u2081\u22c5a\u2082\u22c5a\u2083 + a\u2081\u22c5a\u2083\u22c5a\u2084 + a\u2081 + a\u2083\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                 a\u2080\u22c5a\u2081\u22c5a\u2082\u22c5a\u2084 + a\u2080\u22c5a\u2084 + a\u2081\u22c5a\u2082 + a\u2081\u22c5a\u2084 + 1                 "
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cancel(list_to_frac(continued_frac())) == orig_frac"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}